High Impedance Granular Aluminum Ring Resonators

Abstract

Superconducting inductors with impedance surpassing the resistance quantum, i.e., superinductors, are important for quantum technologies because they enable the development of protected qubits, enhance coupling to systems with small electric dipole moments, and facilitate the study of phase-slip physics. We demonstrate superinductors with densely packed meandered traces of granular aluminum (grAl) with inductances up to $4\,μ\mathrm{H}$, achieving impedances exceeding $100\,\mathrm{k}Ω$ in the $4-8\,\mathrm{GHz}$ range. Ring resonators made with grAl meandered superinductors exhibit quality factors on the order of $10^5$ in the single-photon regime and low non-linearity on the order of tens of $\mathrm{Hz}$. Depending on the grAl resistivity, at $10\,\mathrm{Hz}$, we measure frequency noise spectral densities in the range of $10^2$ to $10^3\,\mathrm{Hz}/\sqrt{\mathrm{Hz}}$. In some devices, in the single-photon regime, we observe a positive Kerr coefficient of unknown origin. Using more complex fabrication, the devices could be released from the substrate, either freestanding or suspended on a membrane, thereby further improving their impedance by a factor of three.

Figures (9)

• Figure 1: Granular Aluminum (grAl) Ring Resonators Design. (a) Energy-filtered transmission electron microscopy images of high resistivity grAl (top, $\approx25000µΩcm)$ and low resistivity grAl (bottom, $\approx120µΩcm$). The energy filter at $15eV$ highlights the aluminum volume plasmon. The viewing angle is tilted by 72° to the deposition direction for both images. The right-hand schematic is there to aid interpretation of the images. The grAl was deposited on carbon laces visible at the bottom of the films. The dark particles are gold nanoparticles (Au NPs) introduced as fiducial markers for tracking the sample during rotation in tomography experiments. (b) Schematic of granular aluminum, illustrating aluminum grains (gray) encircled by aluminum oxide barriers. This microscopic configuration forms an effective array of Josephson junctions, which gives high kinetic inductance and tunable non-linearity. (c) Design of the ring resonator and its equivalent circuit. The densely packed meander trace maximizes the ratio of inductance to self-capacitance. The resonator's symmetric design results in two degenerate modes, each involving one effective capacitor shunted by two parallel inductors. (d) Rendering of the electric field distribution to illustrate the charge distribution within the resonator. The modes are similar to dual modes in Aluminum ring resonators in Ref. minevPlanar2013.
• Figure 2: Ring Resonators in CPW Architecture. (a) Three-dimensional rendering of the chip layout on a sapphire substrate featuring multiple CPWs, enabling the measurement of various devices, and a surrounding ground plane to suppress parasitic modes and ensure a uniform magnetic field. The ground plane mostly consists of silver, which avoids flux trapping and damps microwave slot modes and phonons henriquesPhonon2019. The central line of the CPW and the regions close to it consist of thin films of 15nm niobium capped by 5nm of aluminum. The $5\,\mathrm{nm}$ Al capping layer is used to enable clean galvanic contact with the ground in overlapping regions, leveraging the argon milling recipe introduced and tested in Ref. grunhauptArgonIonBeam2017. (b) Optical false colour image of the end section of the CPW coupled to several ring resonators. The central line splits into either rectangular coupling strips or (c) a semicircle at its end. Both the central line and the ground plane in this region consist of 10µ m wide superconducting strips of Nb + Al (see panel (a)), to minimize flux trapping. (d) Scanning Electron Microscope (SEM) images of a granular aluminum ring resonator, highlighting the densely packed meander traces achieved through a single-step e-beam lithography process followed by lift-off. (e, f) SEM images of the meandering lines at increasing magnification.
• Figure 3: Frequency and Impedance of Granular Aluminum Superinductors. (a) The graph shows the phase response of one of the doublet ring resonances with the frequency separations of approximately 25MHz, pointing to the asymmetry due to the difference in coupling capacitance of the two modes and fabrication imperfections. (b) The frequency spectra of the fabricated and measured granular aluminum ring resonators. Each row corresponds to an inductance per square and the corresponding resistivity as determined from equation \ref{['eq:kinetic_inductance']}. (c) The graph determines the impedance of each ring resonator up to the first fundamental mode. The filled circular markers represent the measured impedance of the fabricated and characterized rings, with the highest impedance recorded at $127k\Omega$ at a resonant frequency of 4.6GHz. These values are obtained using the relation $Z = 2 \pi f_0 L$.
• Figure 4: Measured impedance scaling with geometry and resistivity. Plot of measured $Z$ in panel (a) and $Zf$ in panel (b) as a function of $(r_{\text{in}}L_{\square}/pw)^{1/2}$ and $1/r_{\text{in}}$, respectively, confirming their linear dependence (cf. dashed lines), consistent with Eq. \ref{['EqModelZ']}. The color code in both panels is consistent with Fig. \ref{['fig_impedance_frequency']}. Details for each resonator are given in the supplementary Table \ref{['table3_Design']}.
• Figure 5: Resonators Performance. (a) Noise spectral density $S(f)$ as a function of frequency $f$ for grAl resonators. The spectra were computed from time traces and are shown for four different resistivity values. The data fit the model $S(f) = S_0 + S_1/f^\alpha$, with $\alpha$ ranging from 0.8 to 0.97. The two curves corresponding to resistivity values $\rho = 2500 \, \mu\Omega \cdot cm$ and $\rho = 1200 \,\mu\Omega \cdot cm$ plotted in green and purple, respectively, exhibit behavior that is better described by a Lorentzian model combined with a $1/f^{\alpha}$ dependence at higher frequencies. The inset displays the impedance values corresponding to the frequency noise spectral densities measured at $10Hz$. (b) Intrinsic quality factors $Q_i$ plotted against average photon numbers in the granular aluminum resonators up to the bifurcation threshold $\bar{n}_{\text{max}}$. The average number of photons is given by $\bar{n} = 2Q_cP_{\text{cold}}/\omega_0^2$, where $P_{\text{cold}}$ is the incident microwave power at the device grunhauptgralLoss2018. The shaded intervals indicate the uncertainty range due to Fano interference riegerFano2023. For better visibility, we do not show this uncertainty range for the data set in blue, as it approximately corresponds to an order of magnitude. The inset presents impedance values derived from single-photon $Q_i$ measurements. (c) The shift in resonance frequency as a function of the average photon number for four different rings with varying resistivity is shown. The self-Kerr coefficient $K_{11}$ was determined by fitting the frequency shift data to a linear model. Higher resistivity grAl samples exhibit higher $K_{11}$, consistent with theoretical expectations that $K_{11}$ is proportional to $\rho/V$, where $\rho$ is resistivity and $V$ is the volume maleevaCircuit2018. (d) Anomalous shifts in resonance frequency with photon number. In some ring resonators, we observe an exponential shift to higher frequency at low photon numbers. In the top two panels, the purple data points show the two modes of the same ring resonator measured during two different cooldowns: In the first, we observe the exponential shift to a higher frequency for the higher frequency mode, while the lower frequency mode almost symmetrically shifts, but to a lower frequency. This occurs before the usual Kerr non-linearity at higher photon numbers documented in (c). In the second cooldown, the exponential shifts are no longer observed. In the bottom left panel, a similar anomalous shift is observed for a resonator with higher resistivity grAl. Next to it, we show an example of a high resistivity grAl resonator that does not show this anomalous shift.